Mathematical Modeling and Experimental Investigation of the Dynamic Response for an Annular Circular Plate Made of Glass/Polyester Composite Under Different Boundary Conditions

Fiber-reinforced elastic laminated composites are extensively used in several domains owing to their high specific stiffness and strength and low specific density. Several studies were performed to ascertain the factors that affect the composite plates' dynamic properties. This study aims to derive a mathematical model for the dynamic response of the processed composite material in the form of an annular circular shape made of polyester/E-glass composite. The mathematical model was developed based on modified classical annular circular plate theory under dynamic loading, and all its formulas were solved using MATLAB 2023. The mathematical model was also verified with real experimental work involving the vibration test of the fabricated composite plate. The composite plate was processed by reinforcing the polyester matrix with E-glass fibers with a 50% volume fraction each by using the handy lay-up method. After fabrication, the composite plate was tested with a universal vibration tester, where the plate was impacted and released to free vibration, and the deflection was measured experimentally to compare it with the theoretical value calculated from the derived model. The plate was tested under two boundary conditions, namely, simply and built-in supported. The findings show good agreement between theoretical and experimental plate deflections at different angles, particularly at built-in supported boundary conditions. Also, a higher natural frequency was recorded at this condition compared to others, and this may be ascribed to the higher shear stresses involved due to large moments at the ends along with supporting. Meanwhile, the real experimental spectrum of the built-in condition was higher than others, as the sig view curve revealed.


Introduction
Fiber-strengthened elastic laminated composites are increasingly being adopted by aerospace and other industries owing to the material's superior specifc strength, stifness, and low specifc density [1].Composites are shaped as annular circular plates and are used for aerospace applications, submarines, pressure containers, and roof domes [2].Te annular laminated composite is utilized in many complex industrial structures.Hence, the unconstrained vibration of composite plates having a circular hole cap used in high-pressure pump tankers is a critical issue from the fundamental frequency standpoint.Terefore, studying the vibration behavior of annular composite plates is meaningful and fruitful in terms of noise reduction and impact behavior.Numerous investigations were conducted to develop new approaches or improve the available ones to solve vibration problems of mechanical structures.Tey involve classical plate theory (CPT), zigzag theory, 3D elastic theory, layerwise theory, frst-order shear deformation theory (FSDT), and higher order shear deformation theory (HSDT), and others.
In the manufacturing of the structure of air space frames such as the tip of the front of the fuselage airplane, the rockets or other space ships are needed in the design considering the light of the weight with respect to the high strength of the structure and to withstand the impulse dynamic loading that efect on the front face which constructed from diferent size of annular plates.Several studies have been conducted in this feld to develop analytical and numerical solutions for studying the dynamic response of composite materials in diferent geometries and under diferent boundary conditions.
For instance, the force vibration of the composite laminated annular plate under the Winkler foundation was solved by Afsharmanesh, Ghaheri, and Taheri-Behrooz [3] based on CPT and Ritz methods.Rayleigh-Ritz method was adopted by Amabili, Pierandrei, and Frosali [4] to solve free vibration of the riveted annular composite under elastic boundary conditions.Mercan, Baltacıoglu, and Civalek [5] applied FSDT and discrete singular convolution (DSC) methods to fnd the signifcant vibration variables of annular and conical plates made of isotropic FG composite laminated materials.Also, Amabili et al. [6][7][8][9] conducted multiple studies on the forced nonlinear vibration of the composite plate by using third-order shear deformation theory.Khare and Mittal [10] evaluated the free vibration characteristics of laminated composite annular plates under stressful hydrothermal conditions.Te Green--Lagrange nonlinear kinematics approach was employed to devise the mathematical framework for the laminated annular plate based on the higher order shear deformation concept.
Jin et al. [11] contrasted the two-and three-dimensional (2D vs. 3D) fnite elements (FEs), refned and conventional 2D generalized diferential quadrature (GDQ) techniques along with a precise three-dimensional proposal to evaluate free vibrations concerning single-and multilayered isotropic composite annular composite plates.Amabili, Pierandrei, and Frosali [4] evaluated the free vibration challenges concerning laminated composite stifened circular plates having cutouts based on several boundary criteria based on the FE technique.Arshid and Khorshidvand [12] performed a precise free vibration assessment concerning doubly curved laminated composite annular circular plate based on piezoelectric techniques by integrating the dynamic stifness method (DSM) and the HSDT.
Mercan, Baltacıoglu, and Civalek [5] provided mathematical outcomes concerning laminated composite plate/ shell structures based on the FSDT approach to study shear disfgurement using the DSC technique.Viswanathan et al. [13] evaluated free vibrations related to composite-based spherical shell caps without and with a cutout.Te authors used the FE technique associated with the higher order shear deformation concept.Powmya and Narasimhan [14] evaluated the free vibration properties pertaining to a rotating laminated shell based on Chebyshev collocation.Te outcomes were reported primarily for the impact loading scenario.
Wilson [15] used the FE technique to report the free vibration properties pertaining to a composite-based doubly curved laminate subjected to impact loading.Te analysis relies on the frst-order shear deformation concept pertaining to the laminated doubly curved circular sheet.Katariya, Panda, and Mehar [16] evaluated a practical technique for conceptual modeling and practical validation of modal outcomes pertaining to skewed laminated sandwich builds having a soft epoxy-based core; they assessed the free vibration properties of an orthotropic annular spherical sheet having clamped boundary criteria along all edges.
Bounouara et al. [17] studied the linear, nonlinear, and uniform thermal stresses with constant mechanical loading of anisotropic composite materials on the visco-Pasternak along the plate thickness.Te study involved investigating the infuence of geometric ratio, the ratio of the coefcient of thermal expansion, the coefcient of damping, and foundation variables on the thermomechanical bending behavior of thick plate composite materials.Belbachir et al. [18] employed a refned quasi-3D trigonometric shear deformation method to study the stability and dynamic behaviors of laminated composite plates.Four unknowns were included in the displacement feld, besides considering the thickness efect and involving the mathematical model with the Winkler-Pasternak elastic base.Good accuracy was demonstrated when the model results were compared with other literature.
Based on the reviewed works above, it becomes more complicated when utilizing diferent layers of composite material with diferent volumetric ratios especially in annular circular plates with diferent boundary conditions so that the main performance that gives a special performance for the design of the structure especially in the kind of the transient vibration causes good illustration of the factors that afect the design of the structure.Terefore, the current study aims to present mathematical modeling of free vibration for the orthotropic annual circular plate under impact loading and diferent boundary conditions.Te mathematical model was derived based on the modifed classical annular circular plate theory and validated with a real experimental investigation for the processed glass/ polyester composite material.Te processed plate was subjected to clamped, partially supported, and simply supported boundary conditions.

Deriving the Mathematical Model
Te typical circular plate equation is stated as a function of its coordinates (r and θ) [19] 2 Te Scientifc World Journal Te bending and twist moments for the orthotropic plate are specifed as follows: and Te frst term in equation ( 1) becomes Te second term in equation ( 1) becomes Further, the third term in equation ( 1) is stated as follows: Te value can be substituted concerning the three terms in equation ( 1), leading to the following expression: Te homogeneous solution of equation ( 1) is specifed as follows: where A and B are constants.Te homogeneous solutions (1), ( 2), (8), and ( 9) can be determined for r and θ: Tese expressions are substituted in equation ( 1), and the simplifed expression is as follows: And the specifc solution is as follows: Substitute equation (12) in plate equation ( 1) and determine constant C: Te complementary solution concerning defection is specifed as follows: 4 Te Scientifc World Journal It can be used for the boundary criteria in equation ( 1) to determine constants A and B At r � r 1 � 2.5 cm, θ � θ 1 , w(r, θ) � 0. At r � r 1 � 2.5 cm, θ � θ 2 , w(r, θ) � 0. θ 1 and θ 2 difer for every fank. Ten, where It may be said that the two contact load coordinates are equivalent to the simply supported beam having a point load (q) per unit length based on the dynamic impact concept.Te impact velocity change rate is defned as follows: If we use an identical distance, the target and impactor approach due to local compression at the contact point, leading to the approach velocity being μ If the impact is for an extended duration, the Hertzian law may be used to derive Te Scientifc World Journal where k 1 and K 2 are constants based on impactor and target characteristics.Te fnal equation after diferentiating equation ( 18) and merging with equation (1), followed by substitution of equation ( 19), produces Te two sides of equation ( 21) are multiplied by μ • , and the expression is integrated: where V denotes the approach velocity concerning the two bodies at t � 0 when impact starts; hence, maximum deformation (μ 1 ) occurs at μ • � 0 and is specifed as follows: Substituting of equation ( 23) into equation (19) gives Te potential energy expression uses this q value, leading to energy being a function of initial impact velocity, mass, and plate characteristics and dimensions.Hence, the fundamental frequency is computed using potential energy, and the values provide varying shape factors concerning the natural frequency.
All the above equations have been solved by using MATLAB 2023 by applying the following steps: • For the orthotropic plate, fnd the value of M r , M θ , and M rθ using equation ( 2) as a function of w. • Find the plate rigidity (D r , D θ , and D rθ ) using equation ( 2).• Calculate the frst, second, and third terms of equation ( 1) by substituting the values of M r , M θ , and M rθ , respectively.
• Take the values of these terms and substitute them in the moment equation ( 1) to derive the general equation of motion 3 as a function of w. • Determine the homogeneous defection Wh as a function of r and θ by taking the assumption of sin (r, θ) and cos (r, θ).• Substitute this assumption of Wh in equation ( 1) to fnd its value as a function of (r, θ).• Find the particular solution of Wp as a function of (r, θ).• Apply the boundary conditions to calculate the constants: A, B, C1, C2, and C3, respectively.
• Derive the value of the load q as impact loading taking into account the dynamic impact loading equations ( 17)-( 23).• Te fnal value of q is a function of time from equation (24) which depends on the velocity and type of contact between the two bodies.• Derive the fnal characteristic equation by substituting the value of q to fnd the natural frequency.
Table 1 provides the defnition and unit for all the variables and symbols used in deriving the mathematical model.

Experimental Analysis
In this study, a composite material has been fabricated from a polyester matrix-reinforced woven fberglass and shaped in the form of an annual circular plate.Te composite material was produced using a hand lay-up process employing male and female aluminum molds.Te internal and external diameters of the half-male mold are 20 and 40 cm with 20 mm thickness.
Preparation of the orthotropic plate required the mold surface to be cleaned using acetone, followed by wax coating.Te E-glass fbers were positioned and distributed radially with a 2-cm circular pitch and a 2.5-mm vertical pitch, as depicted in Figure 1.Later, the resin is layered evenly using a brush, followed by the initial mat layer laid using a steel roller over the fabric.Te previous step improves impregnation and wetting; subsequently, the second mat layer is deposited.Tis process is used to lay all fabric layers, and the female mold is used to press the layers.Processing of the composite was performed at room temperature for about 8 hr, followed by removal from the mold to get a useable composite plate, as depicted in Figure 1.

Mechanical Properties.
Te standard mechanical characteristics of constituents of the test samples (E-glass orthotropic fbers and polyester matrix) are provided in Table 2. Te composite shell has 1737.5 kg/m 3 density, 50% volume fraction, 22-GPa elasticity modulus, and 0.27 Poisson ratio.

Vibration Test.
Te fabricated orthotropic composite circular plate was subjected to a vibration test using a universal vibration tester.Te circular plate was tested under two boundary conditions, namely, built and simply supported.Te position of the impact location for both boundary conditions is illustrated in Figure 2. At each boundary condition, the location is impacted down and released to vibrate freely.A Piezo vibration sensor (type MEAS) was attached under the clamping as shown in Figure 3. Te sensor, according to the manufacturer, can generate a voltage of up to 90 volts when it experiences a vibration or impact loading.Te sensor is connected to an oscilloscope (type DS1102E).Te oscilloscope generates a wave-like signal as shown in Figure 4 when the plate is allowed to vibrate freely after impacting and releasing.

Results and Discussions
Tis study aims to derive a mathematical model based on modifed classical annular circular plate theory under dynamic loading and validated with real experimental work to ascertain the parameters that impact the natural frequency corresponding to annular circular composite plates.Hence, the dynamic characteristics should evaluate the high defection value pertaining to the support criteria (boundary conditions).Te change in the composite's natural frequency (either increasing or decreasing) should consider the free vibration duration and the dynamic characteristics of the annular circular composite plate.
Figure 5 depicts the movement of the composite plate against the contact angle pertaining to a simply supported scenario.Te plate defection profle rises exponentially with  Te Scientifc World Journal the contact angle pertaining to the orthotropic annular circular plate.Te error percentages concerning the conceptual and practical outcomes align well since the orthotropic plate has a very high curve degree.Figure 6 shows the defection of the annular circular composite plate against the angle of contact for built-in support.Increasing of contact angle led to a sinusoidal defection since contact loading for several points is minimal or nearly fxed.Te measured natural frequency was varied based on the boundary conditions.For example, the recorded natural frequency for the simply supported was 750 Hz compared with 875 Hz for the built-in boundary conditions.For built-in boundary criteria, shear stress impact rises because of moments at the ends along with the supporting loads.Tis transverse stress leads to a higher natural frequency.Typically, theoretical outcomes align well with practical results; consequently, the numerical is impacted by the change in representation concerning impact loading and the peak potential energy that changes with composite material characteristics.Te Scientifc World Journal

Impact location
In order to highlight the aspects that have a substantial efect on natural frequency increase pertaining to the annular circular composite plate, Figures 7 and 8 provide precise information concerning the increasing response, specifcally for the built-in boundary criteria.In these fgures, it can be noticed that the time response for transmitting the impulse waves for high frequency for the built-in annular circular plate when compared with simply supported conditions.It can be attributed these results of the defections for the annular circular plate to the efect of orientation of the fbers and the volumetric ratio of thecomposite material that led to rise the wave propagation to be more uniform in built-in condition, although the time duration of test for the annular circular composite plate was very short.

. Conclusions
Based on the results obtained, the accuracy and efectiveness of the newly developed mathematical model have been proven, which is capable of calculating the response of a circular plate made of a glass/polyester composite under diferent boundary conditions.Tis mathematical model, which has been verifed with the results of experimental work, has the ability to overcome the difculties faced by other researchers, which is considered useful in the process of designing panels that meet the requirements of the industrial sector on the one hand and the customer on the other hand.Te main goal is to obtain the best possible performance, taking into account optimal safety within a wide range of vibration frequencies.Finally, it can be summarized the main points that were concluded based on the obtained results as follows: • Te mathematical model has been derived successfully to model the free vibration of the processed annular circular composite plate.• Te mathematical model has been verifed efciently with real experiments by subjecting the fabricated composite plate to a vibration test.
• Tere was good agreement between theoretical and experimental values particularly for the built-in supported boundary conditions.• Te measured natural frequency for built-in support was greater due to the increasing shear stress impact as a result of moments at the ends along with supporting loads that lead to higher natural frequency compared to simply supported boundary conditions.• Consequently, the built-in condition generates a higher natural frequency spectrum as the sig view shows.

Table 1 : 2 Figure 1 :
Figure 1: Te fabricated circular composite plate: (a) male mold with distributed E-glass fbers before layering the resin and (b) the fnal part after resin layering.

Figure 2 :Figure 3 :
Figure 2: Te three testing locations of the boundary conditions.

Figure 4 :Figure 5 :
Figure 4: Test the wave that comes from the free vibration of the annular circular plate.

Figure 6 :Figure 7 :
Figure 6: Te defection of the annular circular composite plate against the angle of contact for built-in supported.

Table 2 :
Mechanical characteristics of E-glass fber and polyester resin.